On Hom-F-manifold algebras and quantization

Abstract

The notion of a F -manifold algebras is an algebraic description of a F -manifold. In this paper, we introduce the notion of Hom-F -manifold algebras which is generalisation of F -manifold algebras and Hom-Poisson algebras. We develop the representation theory of Hom-F -manifold algebras and generalize the notion of Hom-pre-Poisson algebras by introducing the Hom-pre-F -manifold algebras which give rise to a Hom-F -manifold algebra through the subadjacent commutative Hom-associative algebra and the subadjacent Hom-Lie algebra. Using O -operators on a Hom-F -manifold algebras we construct a Hom-pre-F -manifold algebras on a module. Then, we study Hom-pre-Lie formal deformations of commutative Hom-associative algebra and we prove that Hom-F -manifold algebras are the corresponding semiclassical limits. Finally, we study Hom-Lie infinitesimal deformations and extension of Hom-pre-Lie n-deformation to Hom-preLie (n + 1)-deformation of a commutative Hom-associative algebra via cohomology theory.